Being embarrassed is certainly one of the more uncomfortable emotions. But, on top of all, do other people really need to see that we are embarrassed? Well we show them, whether we like it or not, through blushing.
How blushing works is understood quite well. Adrenaline is released increasing the heart rate, dilating blood vessels and improving oxygen transport. In humans, the veins of the face (and only the face) react to the adrenalin rush causing blushing.
While the “how” seems to be quite clear, the “why” is not so well understood. There are different theories, the most popular one revolving around the assumption the blushing is a way of convincing people we have understood that we have a done something socially unacceptable. Since blushing is involuntary people seem to be more sympathetic with a blushing person, because they do not expect manipulation.
So the next time you get red in the face, be a little more grateful for this response. I might just have saved your reputation.